Rendiconti di Matematica e delle Sue Applicazioni (Jan 2006)
Approximations of set-valued functions based on the metric average
Abstract
This paper investigates the approximation of set-valued functions with compact images (not necessarily convex), by adaptations of the Schoenberg spline operators and the Bernstein polynomial operators. When replacing the sum between numbers in these operators, by the Minkowski sum between sets, the resulting operators approximate only set valued functions with compact-convex images [10]. To obtain operators which approximate set-valued functions with compact images, we use the well known fact that both types of operators for real-valued functions can be evaluated by repeated binary weighted averages, starting from pairs of function values. Replacing the binary weighted averages between numbers by a binary operation between compact sets, introduced in [1] and termed in [4] the “metric average”, we obtain operators which are defined for set-valued functions. We prove that the Schoenberg operators so defined approximate set-valued functions which are Hölder continuous, while for the Bernstein operators we prove approximation only for Lipschitz continuous set-valued functions with images in R all of the same topology. Examples illustrating the approximation results are presented.
